Alas, the definition they prefer is one of the ones I do not like. They say:

An individual's personal fitness is defined as the number of offspring that she produces that survive to adulthood (Dawkins, 1982; Grafen, 2007b; Hamilton, 1964; Maynard Smith, 1983; also termed neighbour-modulated fitness). From an evolutionary point of view, a behaviour (or action) is social if it has fitness consequences for both the individual that performs that behaviour (the actor) and another individual (the recipient). Hamilton (1964) classified social behaviours according to whether the consequences they entail for the actor and recipient are beneficial (increase personal fitness) or costly (decrease personal fitness) (Table 2). A behaviour which is beneficial to the actor and costly to the recipient (+/-) is selfish, a behaviour which is beneficial to both the actor and the recipient (+/+) is mutually beneficial, a behaviour which is costly to the actor and beneficial to the recipient (-/+) is altruistic, and a behaviour which is costly to both the actor and the recipient (-/-) is spiteful (Hamilton, 1964; Hamilton, 1970; West et al., 2007b

It is rather ironic that they cite Hamilton, 1964 here. Hamilton was pretty clear in saying that it was inclusive fitness he was talking about - not personal fitness. He says so plainly - at the bottom of page 14 of the paper.

I'm not drawing attention to this blunder in order to bash West, El Mouden and Gardner. My point is just that - if a paper by these folks trying to clear up confusion in the area is in this kind of conflict with its primary sources, then the situation is confusing indeed.

If I did use the term "altruism" I would have to constantly define it. My usual strategy would be to put up a page rather like this one, saying what I mean by the term - and then hyperlink to it whenever I needed to use the term. However, in this case, my preferred definition is a minority taste. W. D. Hamilton was apparently on my side of the issue - but that was long, long ago. I can't bring myself to abandon inclusive fitness and adopt personal fitness - that just seems scientifically stupid to me. However, rather than trying to fight this battle, I have mostly decided to just abandon the term "altruism", as being too polluted. Instead, I mostly use the term "cooperation". It doesn't mean the same thing - but it is much, much less confusing.